A153765 McKay-Thompson series of class 15A for the Monster group with a(0) = 4.

1, 4, 8, 22, 42, 70, 155, 246, 421, 722, 1101, 1730, 2761, 4062, 6106, 9040, 13065, 18806, 27081, 37950, 53183, 74290, 102213, 140048, 191612, 258426, 348300, 467484, 622023, 825016, 1090957, 1432290, 1875930, 2448610, 3179136, 4114996

Offset

-1,2

Links

G. C. Greubel, Table of n, a(n) for n = -1..1000

Formula

Expansion of (A(q) + 3 / A(q))^2 in powers of q^2 where A(q) is g.f. for A058624.

Expansion of 1 + A(q) - 1 / A(q) in powers of q where A(q) is g.f. for A153084.

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u + v - 12) * (u^2 + 7*u*v + v^2) - u*v * (u*v - 63).

G.f. is a period 1 Fourier series which satisfies f(-1 / (15 t)) = f(t) where q = exp(2 Pi i t).

a(n) ~ exp(4*Pi*sqrt(n/15)) / (sqrt(2) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 18 2017

Example

T15A = 1/q + 4 + 8*q + 22*q^2 + 42*q^3 + 70*q^4 + 155*q^5 + 246*q^6 + 421*q^7 + ...

Mathematica

QP = QPochhammer; A = QP[q]*(QP[q^5]/(QP[q^3]*QP[q^15])); s = (A + 3*(q/A))^2 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 16 2015, adapted from PARI *)

Prog

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); A = eta(x + A) * eta(x^5 + A) / (eta(x^3 + A) * eta(x^15 + A)); polcoeff( (A + 3 * x / A)^2, n))}
(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); A = (eta(x^3 + A) * eta(x^5 + A) / (eta(x + A) * eta(x^15 + A)))^3 ; polcoeff( A + x - x^2 / A, n))}

Crossrefs

A058508(n) = a(n) unless n = 0. Convolution square of A058625.

Cf. A134783. - R. J. Mathar, Jan 07 2009

Sequence in context: A323584 A332199 A200149 * A003606 A048657 A322284

Adjacent sequences: A153762 A153763 A153764 * A153766 A153767 A153768

Keyword

nonn

Author

Michael Somos, Jan 01 2009

Status

approved